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Ball Triangle Picking

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BallTrianglePickingDistribution

Ball triangle picking is the selection of triples of points (corresponding to vertices of a general triangle) randomly placed inside a ball. n random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], {n, 3}].

The distribution of areas of a triangle with vertices picked at random in a unit ball is illustrated above. The mean triangle area is

A^_=9/(77)pi
(1)

(Buchta and Müller 1984, Finch 2010).

n random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], {n, 3}].

The determination of the probability for obtaining an acute triangle by picking three points at random in the unit disk was generalized by Hall (1982) to the n-dimensional ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals. Let P_n be the probability that three points chosen independently and uniformly from the n-ball form an acute triangle, then

P_(2m+1)=-1/2-2^(2m-1)((2m; m)(4m; 2m))/((4m; m)(6m+1; 2m))+m(2m; m)^22^(2m)sum_(k=0)^(m)((2k; k))/((2m+k; m)(4m+2k; 2m+k))(3m+k+1)/((m+k)(3m+2k+1))
(2)
P_(2m+2)=1/4-3/(2^(2m+4))((4m+4; m+1))/((2m+2; m+1))+(2^(4m))/((2m; m)pi^2)[1/((2m+1)^2(2m; m))+sum_(k=0)^(m)(2^(2k)(3m+k-3))/((2k+1)(2k; k)(2m+k; m)(2m+k+2; 2))].
(3)

These can be combined and written in the slightly messy closed form

P_n=pi^(-3/2){2^(2n-5)(n-1)[Gamma(1/2n)]^4[n_3F^~_2(1,n+1,1/2n+1;1/2(n+3),3/2n+1;1)-2_3F^~_2(1,n,1/2n+1;3/2n,1/2(n+3);1)] -(sqrt(pi)Gamma(2n))/(4^nGamma(3/2n)Gamma(1/2(n+1)))+1},
(4)

where _3F^~_2(a,b,c;d,e;z) is a regularized hypergeometric function.

BallTrianglePicking

The first few are

P_2=4/(pi^2)-1/8 approx 0.280285
(5)
P_3=(33)/(70) approx 0.471429
(6)
P_4=(256)/(45pi^2)+1/(32) approx 0.607655
(7)
P_5=(1415)/(2002) approx 0.706793
(8)
P_6=(2048)/(315pi^2)+(31)/(256) approx 0.779842
(9)
P_7=(231161)/(277134) approx 0.834113
(10)
P_8=(4194304)/(606375pi^2)+(89)/(512) approx 0.874668
(11)
P_9=(9615369)/(10623470) approx 0.905106
(12)

(OEIS A093756 and A093757, OEIS A093758 and A093759, and OEIS A093760 and A093761), plotted above.

The case P_2 corresponds to disk triangle picking.

See also

Cube Triangle Picking, Disk Triangle Picking, Geometric Probability, Obtuse Triangle, Sphere Point Picking

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References

Buchta, C. "A Note on the Volume of a Random Polytope in a Tetrahedron." Ill. J. Math. 30, 653-659, 1986.Buchta, C. and Müller, J. "Random Polytopes in a Ball." J. Appl. Prob. 21, 753-762, 1984.Finch, S. "Random Triangles III." Apr. 30, 2010. https://web.archive.org/web/20160419124628/http://www.people.fas.harvard.edu/~sfinch/csolve/rtg3.pdf.Hall, G. R. "Acute Triangles in the n-Ball." J. Appl. Prob. 19, 712-715, 1982.Sloane, N. J. A. Sequences A093756, A093757, A093758, A093759, A093760, and A093761 in "The On-Line Encyclopedia of Integer Sequences."

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Ball Triangle Picking

Cite this as:

Weisstein, Eric W. "Ball Triangle Picking." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BallTrianglePicking.html

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